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WO2012177365A2 - Prévision de vente au détail à l'aide d'estimation de paramètres - Google Patents

Prévision de vente au détail à l'aide d'estimation de paramètres Download PDF

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Publication number
WO2012177365A2
WO2012177365A2 PCT/US2012/039971 US2012039971W WO2012177365A2 WO 2012177365 A2 WO2012177365 A2 WO 2012177365A2 US 2012039971 W US2012039971 W US 2012039971W WO 2012177365 A2 WO2012177365 A2 WO 2012177365A2
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WIPO (PCT)
Prior art keywords
linear program
dual
dual linear
input data
parameter estimation
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Ceased
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PCT/US2012/039971
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WO2012177365A3 (fr
Inventor
Shivaram Subramanian
Bharath Kumar KRISHNAN
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Oracle International Corp
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Oracle International Corp
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Priority to JP2014516983A priority Critical patent/JP2014520340A/ja
Priority to CN201280031078.9A priority patent/CN103649942B/zh
Priority to EP12802954.3A priority patent/EP2724252A4/fr
Publication of WO2012177365A2 publication Critical patent/WO2012177365A2/fr
Publication of WO2012177365A3 publication Critical patent/WO2012177365A3/fr
Anticipated expiration legal-status Critical
Ceased legal-status Critical Current

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    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06Q—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL OR SUPERVISORY PURPOSES, NOT OTHERWISE PROVIDED FOR
    • G06Q30/00—Commerce
    • G06Q30/02—Marketing; Price estimation or determination; Fundraising
    • G06Q30/0201—Market modelling; Market analysis; Collecting market data
    • G06Q30/0202—Market predictions or forecasting for commercial activities

Definitions

  • One embodiment is directed generally to a computer system, and in particular to a retail forecasting computer system that uses parameter estimation.
  • Parameter estimation provides for the efficient use and analysis of data to aid in mathematically modeling phenomena and the estimation of constants appearing in these models.
  • Much of parameter estimation can be related to four optimization problems: (1) criterion: the choice of the best function to optimize (minimize or maximize); (2) estimation: the optimization of the chosen function; (3) design: optimal design to obtain the best parameter estimates; and (4) modeling: the determination of the mathematical model which best describes the system from which data are measured.
  • Parameter estimation methods typically use the technique of linear regression, also known as "ordinary least-squares" ("OLS").
  • OLS ordinary least-squares
  • Such methods are known to not be “robust” in that they can be vulnerable to outliers in data.
  • the addition of a single outlying data point can significantly affect the value of the estimated parameters.
  • One embodiment is a system that generates a parameter estimation of a plurality of variables.
  • the system receives input data that includes user acceptance criteria and user objectives.
  • the system encodes the input data into a matrix and transforms the input data using a dual linear program.
  • the system solves the dual linear program using a dual simplex method with boxed variables, and recovers parameter values for the parameter estimation.
  • the parameter estimation may be used to provide retail forecasting.
  • FIG. 1 is a block diagram of a computer system that can implement an embodiment of the present invention.
  • Fig. 2 is a flow diagram of the functionality of the retail parameter estimation module of Fig. 1 when generating retail parameter estimation in accordance with one embodiment.
  • One embodiment is a computer system for retail modeling and forecasting using parameter estimation.
  • the system stores input data in a dense matrix format such as a floating point array, creates additional temporary arrays to represent an appropriate dual formulation, and solves the resultant specially structured dual linear program using a revised dual simplex method for boxed- variables to the desired level of optimality.
  • the result is parameter values that are used for retail modeling and forecasting.
  • Parameter estimation methods can be used for retail forecasting. For example, methods can be used for determining the price elasticity of demand for a consumer product, such as if the price of a shirt is reduced 20%, how much will sales increase.
  • Some known parameter estimation methods such as linear regression, also known as “ordinary least- squares” (“OLS”), as discussed above, is vulnerable to outliers in data.
  • Other known methods include “maximum likelihood estimation” (“MLE”).
  • Linear programming has been used for parameter estimation problems because it has been shown to generate robust answers that are relatively immune to data outliers. LP also allows the specification of user-acceptance constraints. However, even the most scalable known LP implementations are too compute-intensive and have not proven to be fully practical, especially when applied to a direct/native LP encoding of such parameter estimation problems involving a large amount of data.
  • Fig. 1 is a block diagram of a computer system 10 that can implement an embodiment of the present invention. Although shown as a single system, the functionality of system 10 can be implemented as a distributed system.
  • System 10 includes a bus 12 or other communication mechanism for communicating information, and a processor 22 coupled to bus 12 for processing information.
  • Processor 22 may be any type of general or specific purpose processor.
  • System 10 further includes a memory 14 for storing information and instructions to be executed by processor 22.
  • Memory 14 can be comprised of any combination of random access memory (“RAM”), read only memory (“ROM”), static storage such as a magnetic or optical disk, or any other type of computer readable media.
  • System 10 further includes a communication device 20, such as a network interface card, to provide access to a network. Therefore, a user may interface with system 10 directly, or remotely through a network or any other method.
  • Computer readable media may be any available media that can be accessed by processor 22 and includes both volatile and nonvolatile media, removable and non-removable media, and communication media.
  • Communication media may include computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media.
  • Processor 22 is further coupled via bus 12 to a display 24, such as a Liquid Crystal Display (“LCD”), for displaying information to a user.
  • a display 24 such as a Liquid Crystal Display (“LCD”)
  • LCD Liquid Crystal Display
  • a keyboard 26 and a cursor control device 28, such as a computer mouse, is further coupled to bus 12 to enable a user to interface with system 10.
  • memory 14 stores software modules that provide functionality when executed by processor 22.
  • the modules include an operating system 15 that provides operating system functionality for system 10.
  • the modules further include a retail parameter estimation module 16 that uses parameter estimation to price, forecast and model retail sales, as disclosed in more detail below.
  • System 10 can be part of a larger system, such as an enterprise resource planning ("ERP") system. Therefore, system 10 will typically include one or more additional functional modules 18 to include the additional functionality.
  • ERP enterprise resource planning
  • a database 17 is coupled to bus 12 to provide centralized storage for modules 16 and 18 and store pricing information, inventory information, etc.
  • Si user-acceptable sign constraint for parameter i (this requires parameter i to take a value that is either non-negative or non-positive);
  • W j relative weight (importance) of observation j
  • b j RHS value of dependent variable (target) in observation j, e.g. sales lift, weekly sales rate, etc.;
  • (Output Data) vak value of parameter (decision variable whose value is to be determined analytically) that satisfies all user acceptance constraints while minimizing the sum of absolute errors between predicted and observed values for the dependent variable, across all observations.
  • a traditional approach to solve the above is to use the ordinary least-squares approach (“OLS”) that minimizes the sum of squared errors given by Min ⁇ j (bj - ⁇ ia ij *val i ) 2 .
  • OLS ordinary least-squares approach
  • QP quadratic programming
  • the error-minimization functional form in a LP is non-differentiable and non- smooth, and thus impacts performance requirements.
  • Solving large LPs typically requires investment in so-called sparse commercial LP solvers (e.g., the IBM ILOG CPLEX Optimizer from IBM Corp., or Gurobi Optimization from Gurobi Corp.).
  • sparse commercial LP solvers e.g., the IBM ILOG CPLEX Optimizer from IBM Corp., or Gurobi Optimization from Gurobi Corp.
  • Embodiments of the present invention utilize efficient LP formulations for parameter estimation derived via a three-step sequence of transformations. Embodiments further use novel metrics and user-acceptance requirements in the form of a variety of user- acceptance constraints and error-minimization objectives. Embodiments exploit the special structure of the resulting LP formations to significantly improve practical convergence. Further, embodiments use a triplet data structure in combination with a quick-sort subroutine that speeds up empirical convergence of the solution approach.
  • Fig. 2 is a flow diagram of the functionality of retail parameter estimation module 16 of Fig. 1 when generating retail parameter estimation in accordance with one embodiment.
  • the functionality of the flow diagram of Fig. 2 is implemented by software stored in memory or other computer readable or tangible medium, and executed by a processor.
  • the functionality may be performed by hardware (e.g., through the use of an application specific integrated circuit ("ASIC"), a programmable gate array (“PGA”), a field programmable gate array (“FPGA”), etc.), or any combination of hardware and software.
  • ASIC application specific integrated circuit
  • PGA programmable gate array
  • FPGA field programmable gate array
  • input data is received, including user acceptance criteria, user objectives and observations and variables of interest.
  • the user acceptance criteria may include the user's intuition on how the model should behave, and hard constraints (e.g., price cuts should never lead to decreased sales) and soft constraints (e.g., front page advertisements tend to do worse than back page advertisements).
  • the input data at 202 can be in the following form in one embodiment:
  • a regressor coefficient matrix i.e., a floating point two-dimensional array whose elements represents elements a '[][]) ⁇ This input is typically in sparse form (i.e., no non-zero elements and their positions are provided).
  • Desired error-minimization objective minimize sum of absolute deviations, minimize maximum deviation, or minimize a combination of the two objectives.
  • the input data is encoded in a dense matrix format (i.e., using standard floating-point arrays).
  • the resultant specially structured dual linear program is solved using a revised dual simplex method for boxed-variables disclosed below to the desired level of optimality.
  • a sorting method to induce rapid convergence for this structure is also deployed.
  • a standard LP formulation is transformed via a three step transformation to generate one of three novel dual LP formulations (referred to as "DP",
  • Transformation Step 1 Original LP Formulation (Minimizing weighted sum of least absolute deviations):
  • vah decision variable j3 ⁇ 4
  • Transformation 1 Convert to a linear program:
  • the resultant LP can be shown be equivalent to the following problem LP:
  • one embodiment uses the theory of duality to generate the dual formulation to this problem as disclosed below.
  • Transformation 2 Compute an equivalent LP using duality theory to generate "DP": DP : Maximize
  • c) DP has a small number (m) of constraints and a large number of ( «) columns via the decision variables %
  • One embodiment solves such parameter estimation problems that are characterized by novel user-acceptance constraints and novel user-specified business objectives. This solution is achieved by recognizing the hidden practical implications of the mathematical properties of the novel transformations in the context of parameter estimation and then finding the best available approach, and, if desired, suitably modifying it via novel enhancements to speed it up.
  • One embodiment recognizes the fact that DP has a small number of rows. Therefore, a revised simplex method can be used to solve this problem. Since the number of rows is small, it allows a small working basis to be employed that is defined by an m X m square matrix. Since the working basis is small, simple and direct arithmetic operations can be performed to obtain the inverse of that matrix and other row and column operations required in a simplex implementation, without requiring sophisticated libraries. Therefore, even though the original primal problem includes millions of rows and columns,
  • embodiments adopt a far simpler dense-matrix approach as opposed to the far more sophisticated sparse-matrix methods employed in commercial packages, which requires either a massive amount of incremental expense to purchase, or significant engineering work to complete (e.g., a factor of 20 more effort).
  • Embodiments use the dual simplex method.
  • the dual simplex method is an instance of a simple method for linear program where dual feasibility is always maintained. Therefore, embodiments satisfy most of the user-acceptance constraints at every intermediate step of the method.
  • Embodiments combine the concepts of the disclosed DP linear programming formulation, the revised simplex implementation, the dual simplex method, and efficient and implicit box- variable handling.
  • This combined approach utilizes a simple version of the revised dual-simplex method with box-variables (“RDSM-BV") that will converge quickly.
  • RDSM-BV box-variables
  • the known "long-step” approach (disclosed in, for example, Koberstein, A., "Progress in the dual simplex algorithm for solving large scale LP problems: techniques for a fast and stable implementation", Journal of Computational Optimization and
  • the long-step approach admits an efficient calculation of the bulk of the boxed variables.
  • One key step in the long-step algorithm for RDSM-BV is to compute a score for each boxed-variable. This allows embodiments to decide whether a box-variable will be retained at its current value, moved to its upper bound (w), or moved to its lower bound (-w).
  • Embodiments speed up the basic long-step algorithm by adopting as an intermediate step a specialized triplet data structure for improving efficiency, and a quick-sort algorithm based on an array of these triplets for improving effectiveness for parameter estimation problems.
  • This triplet-quick sort combination is utilized to efficiently compute the most effective order of box-variables such that it empirically maximizes the number of bound swaps for the class of parameter estimation problems addressed.
  • Embodiments use a novel modification of the general steps of the algorithm disclosed in Koberstein, A. "The Dual Simplex Method, Techniques for a fast and stable implementation", PhD Dissertation, Paderborn, Germany, 2005, Chapter 3, Algorithms 4-7, the disclosure of which is herein incorporated by reference.
  • embodiments implement a bound- flip ratio test step ("BFRT") to speed up the class of problems directed to retail forecasting/modeling.
  • BFRT bound- flip ratio test step
  • the embodiments focus strongly on efficiently maximizing the number of bounds that can be flipped at every step of the dual simplex implementation, rather than achieve the maximum improvement in the objective function value.
  • Embodiments create and store an array (referred to as "RCvars") of triplets (i.e., record three coefficients) for each non-basic variable whose bounds can be flipped from its lower to its upper bound or vice-versa.
  • the three coefficients are stored as shown below: Coefficient 1: +k or -k , where k is the non-basic variable index. If the non-basic variable is currently at its lower bound, -k, and +k are stored otherwise.
  • Ratio[A:] The ratio of the reduced cost of the variable to its modified pivot value given by B ⁇ a k .
  • Step 1 Feasibility Test: If( any non-basic variable is at its lower bound (LB) and it's reduced cost (rc) value ⁇ - tolerance) OR
  • Step 2 Check if var at LB can be improved by flipping to its UB
  • nb var is at LB and tolerance:
  • Step 3 Check if var at UB can be improved by flipping to its LB
  • Step 4 Employ any standard sorting algorithm (e.g., the built-in quick-sort algorithm available in "Java” from Oracle Corp.) to sort the triplets stored in RCvars in ascending order (increasing values) of their ratios (coefficient 2). By doing this, the smallest ratios are processed earlier, which in turn flips a maximal number of bounds as can be seen below. Further, the use of this particular triplet structure allows the efficient retrieval of all the relevant information required directly from runtime -memory, without the need to perform additional computations or recalculations.
  • any standard sorting algorithm e.g., the built-in quick-sort algorithm available in "Java” from Oracle Corp.
  • step (2a) As can be observed in step (2a), the smaller the reduction in the current_slope, the lesser the possibility of it becoming negative (thereby terminating the iterations). In practice, it is observed that 75-90% of the boxed variables are flipped in any single long step iteration, which greatly speeds up convergence, and in the end, no more than 10-15 iterations of the dual simplex method are required before optimality is reached.
  • Embodiments employ a novel specific triplet structure to record candidate data in conjunction with the reordering technique of the candidates and subsequent retrieval of the results.
  • DP2 Minimizing maximum error in predicted versus observed value of dependent variable (target) among all observations (i.e., minimizing error on worst-case prediction).
  • Embodiments use the following novel dual formulation (DP2):
  • Additional user-acceptance constraints that can be handled by embodiments include user-imposed lower and upper bounds on parameter values, and a penalty on an absolute deviation from a user-preferred (desired) input parameter value. These constraints are handled using data transformation techniques to convert the constraints to an equivalent lower and upper bounds requirement.
  • embodiments include a series of mathematical transformations to generate an LP formulation for a parameter estimation problem.
  • the LP formulation can be used to estimate parameters for a forecasting model that predicts retail sales resulting from a promotional offer, but can also be used outside of the retail environment.
  • the LP formulations in accordance to embodiments have a special structure that admits a scalable algorithm implementation for robust estimation of parameters while satisfying user-acceptance constraints/requirements and has the further benefit that they converge rapidly in practice.
  • the LP formulations allow for the simultaneous consideration of multiple "goodness of fit" metrics while estimating parameters. Specifically, the average absolute deviation can be minimized while also limiting the maximum absolute deviation with a user-specified value.
  • the user-acceptance requirements are fully integrated into the parameter estimation mathematical formulation and optimized using single call to the LP solver. Therefore, there is no need to sequentially perform parameter estimation, check whether the solution satisfies the user-acceptance requirements, adjust the parameters, re-estimate, etc., as is required with some prior art solutions
  • Embodiments allows advanced modeling tasks such as variable selection

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Abstract

Selon l'invention, un système génère une estimation de paramètres d'une pluralité de variables. Le système reçoit des données d'entrée qui comprennent des critères d'acceptation d'utilisateur et des objectifs d'utilisateur. Le système code les données d'entrée en une matrice et transforme les données d'entrée à l'aide d'un programme linéaire dual. Le système résout ensuite le programme linéaire dual à l'aide d'un procédé duale du Simplexe avec des variables encadrées, et récupère des valeurs de paramètres pour l'estimation de paramètres. L'estimation de paramètres peut être utilisée pour fournir une prévision de vente au détail.
PCT/US2012/039971 2011-06-24 2012-05-30 Prévision de vente au détail à l'aide d'estimation de paramètres Ceased WO2012177365A2 (fr)

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JP2014516983A JP2014520340A (ja) 2011-06-24 2012-05-30 パラメータ推定を用いた小売予測
CN201280031078.9A CN103649942B (zh) 2011-06-24 2012-05-30 利用参数估计的零售预估
EP12802954.3A EP2724252A4 (fr) 2011-06-24 2012-05-30 Prévision de vente au détail à l'aide d'estimation de paramètres

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US13/168,382 US20120330717A1 (en) 2011-06-24 2011-06-24 Retail forecasting using parameter estimation
US13/168,382 2011-06-24

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CN103649942A (zh) 2014-03-19
JP2014520340A (ja) 2014-08-21
WO2012177365A3 (fr) 2013-03-28
CN103649942B (zh) 2016-12-14
EP2724252A4 (fr) 2015-03-04
EP2724252A2 (fr) 2014-04-30
US20120330717A1 (en) 2012-12-27

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